Measures of Variability

Variability

Measure of the spread or dispersion of a set of data

4 main measures of variability– Range– Interquartile range– Variance– Standard deviation

Range

Difference between high and low scores

Provides estimate of variabilityVery unstable

– Changes radically if extreme scores are present in the data set

Interquartile Range (IQR)

Difference between the raw scores at the 75th and 25th percentile points– IQR = Q3 - Q1

More interested in the middle scores than in the extreme scores– Only considers middle half of data set– Not affected by scores at extreme ends

Variance

Determines the distance of each raw score from the mean of the data

Deviation– d

Variance

Average of the squared deviations

€

V =∑d2

N

Standard Deviation

Squaring of deviation scores can cause problems

Need to bring variance into line with other variability measures

Standard deviation– A measure of the spread of the values in a

parametric data set standardized to the scale of the unit of measurement of the data

– Square root of the average of the squared deviations from the mean

Standard Deviation

Standard Deviation of a Population

Definition method

Can be cumbersome and lengthy procedure– Especially when N is large– Increased probability of mathematical error

€

σ =∑d2

N

Raw Score Method

Algebraic manipulation of definition method to allow for the use of raw scores

Simpler to apply– Squaring and

summing of raw scores

€

σ =∑X 2

N− X

2

Calculating Standard Deviation for a Sample

Sample is drawn from the population– Sample smaller than population– Sample should be representative of the

populationVariability of sample is rarely as large

as that of the populationCorrection factor needed in equation

is needed to estimate the SD for the population from the sample

Calculating Standard Deviation for a Sample

Equations already presented assume entire population has been sampled– Error occurs when using these

equations for a sampleDegrees of freedom

– Number of values in a data set that are free to vary

– Correction factorN-1

Equations

Definition formula

Raw score method

€

SD =∑d2

N −1

€

SD =∑X 2 −

∑X( )2

NN −1